# Modeling for Field-Effect Transistors

## Abstract

The transistor industry is vast, in its manufacturing aspect, in the wide use of its products, and in the research it generates both applied and fundamental. In universities most of this work is done in physics and electrical engineering departments and little of its mathematical requirements have been taken up by math faculty. Yet there has been extensive work on modeling, analysis and computer algorithms, and there remain many open and significant problems. In this talk I shall describe some of the modeling, give a brief introduction to the equations governing current flow in a device, review some of the analytic and numerical approaches to the solution of these equations, and refer to some specialized problems where analysis is useful.^{1} In so doing I hope to pique the curiosity of some of my audience into taking a longer look at the rich phenomena in this field.

## Keywords

Current Flow Parameter Extraction Multiple Steady State MOSFET Model Claremont Graduate School## Preview

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## References

- [1]Sze, S. M.: Physics of Semiconductor Devices. John Wiley
*-*Sons, New York, Second Edition, 1981.Google Scholar - [2]Mock, M.S.: Analysis of Mathematical Models of Semiconductor Devices. Dublin: Boole Press 1983.zbMATHGoogle Scholar
- [3]Selberherr, Siegfried: Analysis and Simulation of Semiconductor Devices. Springer-Verlag Wien, New York 1984.Google Scholar
- [4]Smith, R. A.: Semiconductors. Cambridge: Cambridge University Press 1978.Google Scholar
- [5]Markowich, P.A.: The Stationary Semiconductor Device Equations. Springer-Verlag, Wien-New York 1986.Google Scholar
- [6]Shockley, W.: Electrons and Holes in Semiconductors. Van Nostrand, New York 1950.Google Scholar
- [7]Van Roosbroeck, W.: Theory of flow of electrons and holes in germanium and other semiconductors. Bell System Tech. J., 20 (1950) pp. 560–607.Google Scholar
- [8]Bank, R., Jerome, J., and Rose, D.J.: Analytical and numerical aspects of semiconductor device modeling. Computing Methods in Applied Sciences and Engineering V (R. Glowinski and J. Lions, Eds.), North Holland Publishing, Amsterdam (1982), pp. 593–597.Google Scholar
- [9]Jerome, J.: Consistency of Semiconductor Modeling: an Existence/Stability Analysis for the Stationary Van Roosbroeck System. SIAM J. Appl. Math., Vol. 45, No. 4, 1985 pp. 565–590.MathSciNetzbMATHGoogle Scholar
- [10]Seidman, T.: Steady state solutions of diffusion reaction systems with electrostatic convection. Nonlinear Anal., 4 (1980), pp. 623–637.MathSciNetzbMATHCrossRefGoogle Scholar
- [ll]Bank, R., Rose, D.J., and Fichtner, W.: Numerical methods for semiconductor simulation. SIAM J. Stat. Sci. Comp., 4 (1983), pp. 416–435.MathSciNetzbMATHCrossRefGoogle Scholar
- See also: Special Issue on Numerical Simulation of VLSI Devices. IEEE Trans. Electron Devices, Vol. ED-32 (1985).Google Scholar
- [12]Gummel, H.K.: A self-consistent iterative scheme for one-dimensional steady state transistor calculations. IEEE Trans. Electron Devices, Vol. ED-11, (1964), pp. 455–465.Google Scholar
- [13]Please, C.P.: An Analysis of Semiconductor P-N Junctions. IMA Jour. Appl. Math. (1982) 28, pp. 301–318.CrossRefGoogle Scholar
- [14]Markowich, P.A., Ringhofer, C.A.: A Singularly Perturbed Boundary Value Problem Modelling a Semiconductor Device. SIAM J. Appl. Math. Vol. 44, No. 2, (1984), pp. 231–256.MathSciNetzbMATHGoogle Scholar
- [15]Ringhofer, C.: An Asymptotic Analysis of a Transient p-n Junction Model. SIAM J. Appl. Math., Vol. 47, No. 3, (1987), pp. 624–642.MathSciNetzbMATHGoogle Scholar
- [16]Cimatti, G.: On the Shape of the Region of Depletion in a P-N Junction. Bollettino U.M.I. (5) 18-B (1981), pp. 393–409.Google Scholar
- [17]Rubinstein, I.: Multiple Steady States in One-Dimensional Electrodiffusion with Local Electroneutrality. SIAM J. Appl. Math., Vol. 47, No. 5, (1987), pp. 1076–1093.MathSciNetGoogle Scholar
- [18]Pao, H.C., Sah, C.T.: Effects of Diffusion Current on Characteristics of Metal-Oxide (insulator)-Semiconductor Transistors. Solid-State Electronics, Vol. 9 (1966) pp. 927–937.CrossRefGoogle Scholar
- [19]Brews, J.: A Charge-Sheet Model of the MOSFET. Solid-State Electronics, 21, (1978) pp. 345–355.CrossRefGoogle Scholar
- [20]Ihantola, H.K., Moll, J.L.: Design Theory of a Surface Field-Effect Transistor. Solid-State Electronics, 7, (1964) pp. 423–430.CrossRefGoogle Scholar
- [21]Pierret, R.F., Shields, J.A.: Simplified Long-Channel MOSFET Theory. Solid-State Electronics, Vol. 26 (1983) pp. 143–147.CrossRefGoogle Scholar
- [22]Van De Wiele, F.: A Long-Channel MOSFET Model. Solid State Electronics, Vol. 22, (1979), pp. 991–997.CrossRefGoogle Scholar
- [23]Ward, M., Odeh, F., Cohen, D.S.: Asymptotic Methods for MOSFET Modeling. Accepted by SIAM J. Appl. Math.Google Scholar
- [24]Berger, H.H.: Models for Contacts to Planar Devices. Solid-State Electronics, Vol. 15, (1972), p. 145.CrossRefGoogle Scholar
- [25]Loh, W.M., Swirhern, S.E., Schoeyer, T.A., Swandon, K.C. Saraswat: Modeling and Measurement of Contact Resistances. IEEE Trans. Electron Devices, Vol. ED-34, No. 3, (1987), pp. 512–524.CrossRefGoogle Scholar
- [26]Cumberbatch, E., Fang, W.: Three-Dimensional Modelling for Contact Resistance of Current Flow into a Source/Drain Region. Claremont Graduate School, Math Department, Preprint. 1988.Google Scholar
- [27]Gribben, R.J., Martelli, M., Rykken, C., Meiser, V., Turner, G., Wang. Q.: Parameter Extraction and Transistor Model. Mathematics Clinic, Claremont Graduate School, Final Report, 1985.Google Scholar
- [28]Gribben, R.J., Martelli, M.: Optimal parameter extraction of the Brews charge-sheet MOSFET model. Math. Engng. hid., Vol 1, No. 2, 1987, pp. 155–168.zbMATHGoogle Scholar
- [29]Andersson, G., Allen, D., Fleishman, R., Hamza, H., Lacey, S., Larsson, K., Panagiotacopulos, D., Velasco-Hernandez, J.: Parameter Extraction from a Nonlinear MOSFET Model. Mathematics Clinic, Claremont Graduate School, Final Report, 1988.Google Scholar